Optimal. Leaf size=40 \[ \frac{c^2}{a^3 x^2}-\frac{c^2}{3 a^4 x^3}+\frac{2 c^2 \log (x)}{a}+c^2 (-x) \]
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Rubi [A] time = 0.116405, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6157, 6150, 75} \[ \frac{c^2}{a^3 x^2}-\frac{c^2}{3 a^4 x^3}+\frac{2 c^2 \log (x)}{a}+c^2 (-x) \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6150
Rule 75
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^2 \, dx &=\frac{c^2 \int \frac{e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac{c^2 \int \frac{(1-a x)^3 (1+a x)}{x^4} \, dx}{a^4}\\ &=\frac{c^2 \int \left (-a^4+\frac{1}{x^4}-\frac{2 a}{x^3}+\frac{2 a^3}{x}\right ) \, dx}{a^4}\\ &=-\frac{c^2}{3 a^4 x^3}+\frac{c^2}{a^3 x^2}-c^2 x+\frac{2 c^2 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.015661, size = 40, normalized size = 1. \[ \frac{c^2}{a^3 x^2}-\frac{c^2}{3 a^4 x^3}+\frac{2 c^2 \log (x)}{a}+c^2 (-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 39, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{3\,{a}^{4}{x}^{3}}}+{\frac{{c}^{2}}{{x}^{2}{a}^{3}}}-x{c}^{2}+2\,{\frac{{c}^{2}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959533, size = 51, normalized size = 1.27 \begin{align*} -c^{2} x + \frac{2 \, c^{2} \log \left (x\right )}{a} + \frac{3 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73587, size = 99, normalized size = 2.48 \begin{align*} -\frac{3 \, a^{4} c^{2} x^{4} - 6 \, a^{3} c^{2} x^{3} \log \left (x\right ) - 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.93487, size = 39, normalized size = 0.98 \begin{align*} \frac{- a^{4} c^{2} x + 2 a^{3} c^{2} \log{\left (x \right )} + \frac{3 a c^{2} x - c^{2}}{3 x^{3}}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34723, size = 151, normalized size = 3.78 \begin{align*} -\frac{2 \, c^{2} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{2} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (3 \, c^{2} - \frac{5 \, c^{2}}{a x + 1} - \frac{3 \, c^{2}}{{\left (a x + 1\right )}^{2}} + \frac{6 \, c^{2}}{{\left (a x + 1\right )}^{3}}\right )}{\left (a x + 1\right )}}{3 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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